"It should be generally known that Austria does not have an external EU border and is therefore not the first safe country that these people (migrants) set foot in," Johanna Mikl-Leitner told the Austria Press Agency (APA).
"If everybody stuck to the content of the letter (from migration commissioner Dimitris Avramopoulos), then Austria would not have a problem with it. But the letter was clearly sent to the wrong address," Mikl-Leitner said.
Instead, the complaints should be sent to safe countries that the migrants pass through on their way to Austria, she said, in a reference in particular to Greece, the main entry point in the European Union for migrants.
In 2015, over one million people reached Europe's shores -- nearly half of them Syrians fleeing the civil war -- causing the bloc a major political headache.
Austria last year took in 90,000 asylum seekers, making it one of the highest recipients in the EU on a per-capita basis, while almost 10 times that number passed through, mostly to Germany and Sweden.
Faced with a resurgent far-right opposition topping opinion polls, Austria's centrist government this week imposed the new cap and said only 3,200 migrants could pass through per day.
Border controls are being tightened and the government wants only 37,500 asylum claims this year.
On Thursday, amid widespread criticism of Austria, Avramopoulos sent a letter to Mikl-Leiter calling the cap "plainly incompatible with Austria's obligations under European and international law."
The measures have also raised fears that when migrant numbers spike as expected again in the coming months as spring arrives, there will be a dangerous backlog of people along the Balkans route from Greece northwards.
New asylum claims
On Friday, when the new restrictions came into effect, no single migrant entered Austria, due to bad weather.
On Saturday 396 people, many of them families and including some elderly people, entered at the main Spielfeld border crossing point with Slovenia in southern Austria, police said.
But of these only 12 applied for asylum, the others saying they wished to travel onwards to Germany and transported in army buses to migrant centres, police spokesman Wolfgang Braunsar said.
Vienna says its unilateral moves are necessary because a German-backed EU plan agreed in November for Turkey to stem the flow of migrants leaving its shores for Greece is not yet working.
The EU and Turkey are due to hold a summit on March 6 to seek to firm up their agreement, which would see migrants flown directly from Turkey and shared -- in theory -- around certain members of the bloc.
In return for sealing its borders, Turkey would receive several billion euros in aid and other sweeteners including reinvigorating its drive for EU membership and easing visa restrictions.